Comparison with the theory

About theory

What we want to do here is just to introduce the basis to be able to read the two diagrams shown after.

Comolet's diagram

The first one is called the Comolet's diagram. It will give us the final velocity of the bubble, thanks to its diameter and an adimensionnal number called the Morton.
Now, let us explain the formulae that we need right there to use this diagram.

First of all, V+ and d+ are not directly the speed and the diameter of the bubble.
You may wonder what are their expressions:


You can also see that there is another number that we have to take into account. It is called the Morton number. Physically speaking, this number is constructed with the only properties of the fluid.
We can write it as Mo=[gravity x viscosity4]/[interfacial tension3 x gravity2]
Another clearer expression for this Morton number is:

Comolet's Diagramm
- Comolet's diagram -

If you want the equations, here they are:

Stokes relationif d+ < 2Mo1/6

Wallis relationif 2 Mo-1/8 <d+ <3,8 Mo1/14

Comolet relationif d+ >3,8 Mo1/14

Now, it is very simple to read the speed V+ as soon as you have calculated the Morton's number and the diameter d+.
You choose your diameter on the horizontal axis, and then, you follow the vertical axis until you meet the iso Morton curve corresponding to the Morton's number you have previously calculated! Easy, isn't it!

Let us demonstrate. We will do the calculation for the bubble which diameter is 6 cm.
rho#delta rho=1000kg/m3
Nu=10-6 m²/s
Sigma=0,074 N/m

D=6cm: And now, we find that d+ = 21,8 and Mo=2,42.10-11.

So, if you report those numbers on the previous graphic, you will get a V+ equal to something around 3 or 4.
You can get a more precise value for V+ with the formulae given: We see that d+>0,66=d+ >3,8 Mo1/14 so we use the Comolet relation. The result is V+ = 3,34.
Then you invert the formula to get Vinfinite thanks to V+ and you find that: Vinfinite#3,6m/s.

You can also say, when regarding the diagram, that the shape of the bubble is probably something between an ellipse and a parabola.

D=1cm: d+=3,63and this is once again the comolet relation that we have to use.

It gives us V+=1,56, so Vinfinite=1,68m/s.

Diagram Re(Bo,Mo)

The second one gives us the same kind of results, but you have to calculate another number called the Bond number. It represents the relative importance of Gravity compared to the one of interfacial tension.

Here is the expression for the Bond number: 
Now here is the diagram:

Clift's Diagramm
- Diagram Re(Bo,Mo) -

So, if we take back our two previous examples, we find that the Bond number is equal to:

D=6cm: Bo=486
D=1cm: Bo=13,51

We can notice that the bigger the bubble is, the more important the gravity forces is. As a matter of fact, Archimed's force grows up with the size of the bubble, thus explaining this phenomena.

So, now that we have calculated the Bond number we can easily find out the right position on the diagram and then determine the Reynolds number that corresponds to our case. The velocity is then known through the Reynolds number!
Here, we will just look at the shape of the bubble, considering the fact that its velocity has been determined previously with the Comolet's diagram.

You can see that the bubble of 1 cm diameter has a strange form, like previously seen, between an ellipse and a parabola, whereas the bubble of 6 cm diameter is clearly a parabola or an inverted saucer.

The last thing that seems to be interesting, when dealing with adimensionnal numbers, is to define the Weber number, which is defined this way:

Just to mention it, we calculate the Weber for our 1cm diameter bubble.

The Weber number is equal to We=380.

Small Comparison

The aim of this part is to quickly highlight the results that have stroke us.

* First of all, we have been surprised to obtain so good results with Fluent concerning the final velocity of the bubble. As a matter of fact, for the bubble of 1cm diameter, Fluent gives us a velocity of 1,2 to 1,3 m/s, whereas the velocity we were expecting a velocity around 1,7 m/s according to the theory.
So, you can notice that the difference is around 20%, which is quite good!
Nonetheless, our domain is not very long, and the velocity of the bubble is very likely to still evolve after the end of our calculation. Yet, what we can say is that it seems that at the end of our calculation time, the shape of our bubble appears to be very close to the final one. So it is possible that the error between the model calculated with Fluent and the one predicted by the theory is greater than the one given upper.

* Concerning the small bubble (D= 5 mm),  it remains circular, what is predicted by the theory. As a matter of fact, when you look at the diagrams, with such a bubble you are in the zone of spherical bubble : that is what we notice in the presentation of the results.

* As for the biggest one, the error we have made about the boundary conditions have been fatal to our calculations, and considering the fact that one calculation requires a dozen hours, we haven't made another attempt.

* At last the square bubble is a funny case, but we have to admit that it has no physical existence and no other interest than the one of curiosity. But, this put in evidence the fact that interfacial tension is well modelized in Fluent, because the bubble tends to become spherical what is its equilibrium shape, considering the beginning of the simulation.